- Open Access
Some properties of meromorphically multivalent functions
© Xu et al.; licensee Springer. 2012
Received: 18 October 2011
Accepted: 16 April 2012
Published: 16 April 2012
By using the method of differential subordinations, we derive certain properties of meromorphically multivalent functions.
2010 Mathematics Subject Classification: 30C45; 30C55.
Let f(z) and g(z) be analytic in U. Then, we say that f(z) is subordinate to g(z) in U, written f(z) ≺ g(z), if there exists an analytic function w(z) in U, such that |w(z)| ≤ |z| and f(z) = g(w(z)) (z ∈ U). If g(z) is univalent in U, then the subordination f(z) ≺ g(z) is equivalent to f(0) = g(0) and f(U) ⊂ g(U).
Recently, several authors (see, e.g., [1–7]) considered some interesting properties of meromorphically multivalent functions. In the present article, we aim at proving some subordination properties for the class Σ(p).
To derive our results, we need the following lemmas.
2 Main results
Our first result is contained in the following.
The bound β is the best possible for each .
for all z ∈ U because of g(δ) = 0. This proves (2.3).
Hence, we conclude that the bound β is the best possible for each .
Next, we derive the following.
The bound in (2.14) is sharp.
from (2.16), we get the inequality (2.14).
as z → -1. Now the proof of the theorem is complete.
Finally, we discuss the following theorem.
But both (2.23) and (2.24) contradict the assumption (2.17). Therefore, we have Rep(z) > 0 for all z ∈ U. This shows that (2.18) holds true.
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